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Properties

Label 798.2.e.a
Level $798$
Weight $2$
Character orbit 798.e
Analytic conductor $6.372$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [798,2,Mod(265,798)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(798, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("798.265");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 798 = 2 \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 798.e (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.37206208130\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{11} + 2x^{10} + 54x^{8} - 114x^{7} + 120x^{6} + 46x^{5} + 9x^{4} - 4x^{3} + 8x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - q^{3} - q^{4} - \beta_{2} q^{5} - \beta_{4} q^{6} + \beta_{6} q^{7} - \beta_{4} q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - q^{3} - q^{4} - \beta_{2} q^{5} - \beta_{4} q^{6} + \beta_{6} q^{7} - \beta_{4} q^{8} + q^{9} - \beta_{3} q^{10} + ( - \beta_{5} + 1) q^{11} + q^{12} + (\beta_{3} - 2) q^{13} + \beta_{9} q^{14} + \beta_{2} q^{15} + q^{16} + (\beta_{11} + \beta_{7} - \beta_{6}) q^{17} + \beta_{4} q^{18} + (\beta_{9} + \beta_{6} + \cdots - \beta_{2}) q^{19}+ \cdots + ( - \beta_{5} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{3} - 12 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{3} - 12 q^{4} + 12 q^{9} - 4 q^{10} + 12 q^{11} + 12 q^{12} - 20 q^{13} + 4 q^{14} + 12 q^{16} + 8 q^{19} - 12 q^{23} - 12 q^{25} - 12 q^{27} + 4 q^{30} - 24 q^{31} - 12 q^{33} - 4 q^{34} + 4 q^{35} - 12 q^{36} + 20 q^{39} + 4 q^{40} + 16 q^{41} - 4 q^{42} - 12 q^{44} - 12 q^{48} + 4 q^{49} + 20 q^{52} - 4 q^{56} - 8 q^{57} + 8 q^{58} - 40 q^{59} - 12 q^{64} + 12 q^{69} + 24 q^{70} + 12 q^{75} - 8 q^{76} + 8 q^{77} + 12 q^{81} + 8 q^{85} + 16 q^{89} - 4 q^{90} - 24 q^{91} + 12 q^{92} + 24 q^{93} - 44 q^{95} + 60 q^{97} + 16 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{11} + 2x^{10} + 54x^{8} - 114x^{7} + 120x^{6} + 46x^{5} + 9x^{4} - 4x^{3} + 8x^{2} + 4x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1204841 \nu^{11} - 108477401 \nu^{10} + 281608184 \nu^{9} - 368064334 \nu^{8} + 241501496 \nu^{7} + \cdots - 921139278 ) / 62914123 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4581636 \nu^{11} - 8559599 \nu^{10} + 8749533 \nu^{9} - 720282 \nu^{8} + 249994355 \nu^{7} + \cdots + 17917471 ) / 62914123 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4581636 \nu^{11} + 8559599 \nu^{10} - 8749533 \nu^{9} + 720282 \nu^{8} - 249994355 \nu^{7} + \cdots - 17917471 ) / 62914123 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17917471 \nu^{11} - 40416578 \nu^{10} + 44394541 \nu^{9} - 8749533 \nu^{8} + 968263716 \nu^{7} + \cdots + 37014000 ) / 62914123 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 20522401 \nu^{11} - 9775985 \nu^{10} - 35893384 \nu^{9} + 93036208 \nu^{8} + 1072759908 \nu^{7} + \cdots + 230084943 ) / 62914123 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 41974538 \nu^{11} + 142257941 \nu^{10} - 216580592 \nu^{9} + 149544607 \nu^{8} + \cdots + 218567625 ) / 62914123 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 82307490 \nu^{11} - 150483059 \nu^{10} + 117210810 \nu^{9} + 74172527 \nu^{8} + 4388653824 \nu^{7} + \cdots + 393280907 ) / 62914123 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 113238774 \nu^{11} + 252452879 \nu^{10} - 283529283 \nu^{9} + 56703661 \nu^{8} + \cdots - 225912062 ) / 62914123 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 138418019 \nu^{11} - 293957423 \nu^{10} + 301705425 \nu^{9} - 5867022 \nu^{8} + 7435401174 \nu^{7} + \cdots + 377640732 ) / 62914123 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 149885313 \nu^{11} - 353497855 \nu^{10} + 423842543 \nu^{9} - 140795074 \nu^{8} + \cdots + 184817236 ) / 62914123 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 451390185 \nu^{11} + 1040017769 \nu^{10} - 1173084792 \nu^{9} + 251195486 \nu^{8} + \cdots - 760408626 ) / 62914123 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + \beta_{8} - 2\beta_{7} + 2\beta_{6} - 5\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{10} - \beta_{8} - 2\beta_{6} + \beta_{5} + \beta_{4} - 7\beta_{3} + 7\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 14\beta_{10} - 14\beta_{9} - \beta_{7} - \beta_{6} - 5\beta_{5} + 2\beta_{3} - 7\beta _1 - 31 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - \beta_{11} - \beta_{10} + 18 \beta_{9} + 11 \beta_{8} - 18 \beta_{7} + \beta_{6} + 11 \beta_{5} + \cdots + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 49\beta_{11} + 9\beta_{10} + 9\beta_{9} - 29\beta_{8} + 99\beta_{7} - 99\beta_{6} + 213\beta_{4} + 26\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 13 \beta_{11} + 144 \beta_{10} - 16 \beta_{9} + 92 \beta_{8} - 16 \beta_{7} + 144 \beta_{6} + \cdots - 132 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -712\beta_{10} + 712\beta_{9} + 63\beta_{7} + 63\beta_{6} + 196\beta_{5} - 258\beta_{3} + 348\beta _1 + 1510 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 129 \beta_{11} + 182 \beta_{10} - 1109 \beta_{9} - 708 \beta_{8} + 1109 \beta_{7} - 182 \beta_{6} + \cdots - 1119 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2492 \beta_{11} - 397 \beta_{10} - 397 \beta_{9} + 1440 \beta_{8} - 5166 \beta_{7} + \cdots - 2324 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 1162 \beta_{11} - 8413 \beta_{10} + 1798 \beta_{9} - 5285 \beta_{8} + 1798 \beta_{7} - 8413 \beta_{6} + \cdots + 9096 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/798\mathbb{Z}\right)^\times\).

\(n\) \(115\) \(211\) \(533\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
265.1
1.84195 + 1.84195i
1.25342 + 1.25342i
0.382656 + 0.382656i
−0.242887 − 0.242887i
−0.301222 − 0.301222i
−1.93391 − 1.93391i
−1.93391 + 1.93391i
−0.301222 + 0.301222i
−0.242887 + 0.242887i
0.382656 − 0.382656i
1.25342 − 1.25342i
1.84195 − 1.84195i
− 1.00000i −1.00000 −1.00000 − 3.68390i 1.00000i 0.932914 + 2.47582i 1.00000i 1.00000 −3.68390
265.2 − 1.00000i −1.00000 −1.00000 − 2.50684i 1.00000i −2.53963 − 0.741811i 1.00000i 1.00000 −2.50684
265.3 − 1.00000i −1.00000 −1.00000 − 0.765312i 1.00000i −2.29245 + 1.32086i 1.00000i 1.00000 −0.765312
265.4 − 1.00000i −1.00000 −1.00000 0.485775i 1.00000i 2.59339 − 0.523742i 1.00000i 1.00000 0.485775
265.5 − 1.00000i −1.00000 −1.00000 0.602444i 1.00000i −0.307956 − 2.62777i 1.00000i 1.00000 0.602444
265.6 − 1.00000i −1.00000 −1.00000 3.86783i 1.00000i 1.61372 + 2.09664i 1.00000i 1.00000 3.86783
265.7 1.00000i −1.00000 −1.00000 − 3.86783i − 1.00000i 1.61372 − 2.09664i − 1.00000i 1.00000 3.86783
265.8 1.00000i −1.00000 −1.00000 − 0.602444i − 1.00000i −0.307956 + 2.62777i − 1.00000i 1.00000 0.602444
265.9 1.00000i −1.00000 −1.00000 − 0.485775i − 1.00000i 2.59339 + 0.523742i − 1.00000i 1.00000 0.485775
265.10 1.00000i −1.00000 −1.00000 0.765312i − 1.00000i −2.29245 − 1.32086i − 1.00000i 1.00000 −0.765312
265.11 1.00000i −1.00000 −1.00000 2.50684i − 1.00000i −2.53963 + 0.741811i − 1.00000i 1.00000 −2.50684
265.12 1.00000i −1.00000 −1.00000 3.68390i − 1.00000i 0.932914 − 2.47582i − 1.00000i 1.00000 −3.68390
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 265.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 798.2.e.a ✓ 12
3.b odd 2 1 2394.2.e.a 12
7.b odd 2 1 798.2.e.b yes 12
19.b odd 2 1 798.2.e.b yes 12
21.c even 2 1 2394.2.e.b 12
57.d even 2 1 2394.2.e.b 12
133.c even 2 1 inner 798.2.e.a ✓ 12
399.h odd 2 1 2394.2.e.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.e.a ✓ 12 1.a even 1 1 trivial
798.2.e.a ✓ 12 133.c even 2 1 inner
798.2.e.b yes 12 7.b odd 2 1
798.2.e.b yes 12 19.b odd 2 1
2394.2.e.a 12 3.b odd 2 1
2394.2.e.a 12 399.h odd 2 1
2394.2.e.b 12 21.c even 2 1
2394.2.e.b 12 57.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{6} + 10T_{13}^{5} + 24T_{13}^{4} - 16T_{13}^{3} - 92T_{13}^{2} - 32T_{13} + 40 \) acting on \(S_{2}^{\mathrm{new}}(798, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$3$ \( (T + 1)^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 36 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( T^{12} - 2 T^{10} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( (T^{6} - 6 T^{5} + \cdots + 104)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 10 T^{5} + \cdots + 40)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + 136 T^{10} + \cdots + 5125696 \) Copy content Toggle raw display
$19$ \( T^{12} - 8 T^{11} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( (T^{6} + 6 T^{5} + \cdots - 2056)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 152 T^{10} + \cdots + 11505664 \) Copy content Toggle raw display
$31$ \( (T^{6} + 12 T^{5} + \cdots + 5192)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 132 T^{10} + \cdots + 12334144 \) Copy content Toggle raw display
$41$ \( (T^{6} - 8 T^{5} + \cdots + 12224)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 176 T^{4} + \cdots + 4928)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 252 T^{10} + \cdots + 3534400 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 545689600 \) Copy content Toggle raw display
$59$ \( (T^{6} + 20 T^{5} + \cdots - 3520)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 10816000000 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 1894686784 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 125440000 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 8131710976 \) Copy content Toggle raw display
$79$ \( T^{12} + 564 T^{10} + \cdots + 18496 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 203689984 \) Copy content Toggle raw display
$89$ \( (T^{6} - 8 T^{5} + \cdots - 755200)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 30 T^{5} + \cdots - 5656)^{2} \) Copy content Toggle raw display
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